Recently I met with a problem related to Stone-Cech Compactification theorem in Furstenberg's famous paper "non-commuting product."

I try my best to understand Stone-Cech compactification theorem by a not obviously-trivial example. However, when I consider the exponential map in complex plane, I fail to imagine the behaviour for exponential map in compactification space. I guess the compactification space does have complex structure. Can we say more about the further knowledge about the compactification space and the exponential maps in it.

Any comments and remarks will be appreciated.